Một phương pháp chiếu giải bài toán bất đẳng thức biến phân hai cấp

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❍❖⑨◆● ❚❍➚ ❚❍❷❖ ▼❐❚ P❍×❒◆● P❍⑩P ❈❍■➌❯ ●■❷■ ❇⑨■ ❚❖⑩◆ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❍❆■ ❈❻P ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ❍❖⑨◆● ❚❍➚ ❚❍❷❖ ▼❐❚ P❍×❒◆● P❍⑩P ❈❍■➌❯ ●■❷■ ❇⑨■ ❚❖⑩◆ ❇❻❚ ✣➃◆● ❚❍Ù❈ ❇■➌◆ P❍❹◆ ❍❆■ ❈❻P ❈❤✉②➯♥ ♥❣➔♥❤✿ ▼➣ sè✿ ❚❖⑩◆ Ù◆● ❉Ö◆● ✽✹✻✵✶✶✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❈→♥ ❜ë ữợ P ế ◆●❯❨➊◆✱ ◆❿▼ ✷✵✷✵ ✐✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✶ ỵ s ✹ ❈❤÷ì♥❣ ✶✳ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✻ ✶✳✶ ✶✳✷ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✶ ❙ü ❤ë✐ tö ②➳✉✱ ❤ë✐ tö ♠↕♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✷ ❚♦→♥ tû ❝❤✐➳✉ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✸ ◆â♥ ♣❤→♣ t✉②➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✶✳✹ ⑩♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥ ✈➔ t♦→♥ tû ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✶✳✷✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✷ ▼ët ❜➔✐ t♦→♥ tỹ t ữủ ổ t ữợ t tự ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✸ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✹ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✳ ✳ ✳ ✳ ✻ ✳ ✻ ✳ ✼ ✳ ✽ ✳ ✽ ✳ ✶✶ ✳ ✶✶ ✳ ✶✷ ✳ ✶✹ ✳ ✶✻ ❈❤÷ì♥❣ ✷✳ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ✷✷ ✷✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✶✳✷ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✐✈ ✷✳✶✳✸ ✷✳✷ ❚❤✉➟t t♦→♥ ✤↕♦ ❤➔♠ t➠♥❣ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ ✷✳✷✳✶ ▼ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✳ ✳ ✳ ✷✳✷✳✷ ❙ü ❤ë✐ tư ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❝÷í♥❣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✻ ✷✻ ✷✼ ❑➳t ❧✉➟♥ ✸✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✸✸ ✶ ▲í✐ ❝↔♠ ì♥ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✳ ❚r♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ tèt ♥❤➜t ✤➸ tỉ✐ ✤÷đ❝ t❤❛♠ ❣✐❛ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ P❤á♥❣ ✤➔♦ t↕♦✱ ❑❤♦❛ ❚♦→♥ ✲ rữớ qỵ t ổ trü❝ t✐➳♣ ❣✐↔♥❣ ❞↕② ❧ỵ♣ ❈❛♦ ❤å❝ ❚♦→♥ ❑✶✷❆ ✭❦❤â❛ ✷✵✶✽ ✕ ✷✵✷✵✮ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ tự qỵ ụ ữ t tổ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ✣➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ♠ët tổ ổ ữủ sỹ ữợ ✈➔ ❣✐ó♣ ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ P●❙✳❚❙✳ ◆●❯❨➍◆ ❚❍➚ ❚❍❯ ❚❍Õ❨✳ ❚ỉ✐ ①✐♥ tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝ỉ ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ ❝ỉ ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳ ❚ỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ♥❤➜t tỵ✐ ❣✐❛ ✤➻♥❤✱ ỗ ổ ộ trủ ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥ ❍♦➔♥❣ ❚❤à ❚❤↔♦ ỵ H C Ã, à (F, C) NC (x0 ) S(F,C) ❖P(F, C) ❋P(F, C) PC ❇❱■(F, G, C) S(G,C) Ω ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ♠ët t➟♣ ỗ õ rộ H t ổ ữợ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈ỵ✐ →♥❤ ①↕ ❣✐→ F ✈➔ t➟♣ r➔♥❣ ❜✉ë❝ C ♥â♥ ♣❤→♣ t✉②➳♥ ♥❣♦➔✐ ❝õ❛ C t↕✐ x0 t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(F, C) ❜➔✐ t♦→♥ tè✐ ÷✉ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ♣❤➨♣ ❝❤✐➳✉ ♠➯tr✐❝ ❝❤✐➳✉ H ❧➯♥ C ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ t➟♣ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(G, C) t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❇❱■(F, G, C) ✸ ❉❛♥❤ s→❝❤ ❜↔♥❣ ✶✳✶ ✶✳✷ ✶✳✸ ❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ x0 = (5, 5, 5)T ∈ R3 ✱ ❝❤å♥ µ = 1/(k + 2) ✳ ✳ ✳ ✷✶ ❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 2) ✳ ✷✶ ❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 4) ✳ ✷✶ ✹ ▼ð ✤➛✉ ❈❤♦ H ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ ✈ỵ✐ t ổ ữợ Ã, à à C ởt t ỗ õ rộ H✱ ✈➔ →♥❤ ①↕ F : C → H t❤÷í♥❣ ✤÷đ❝ ❣å✐ ❧➔ →♥❤ ①↕ ❣✐→ ✭tr♦♥❣ ♠ët ✈➔✐ tr÷í♥❣ ❤đ♣✱ F ✤✐ tø H tỵ✐ H✮✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✤ì♥ trà✮ tr♦♥❣ H✱ ✈✐➳t t➢t ❱■(F, C)✱ ✤÷đ❝ ♣❤→t ❜✐➸✉ ♥❤÷ s❛✉✿ ❚➻♠ x∗ ∈ C s❛♦ ❝❤♦ F (x∗ ), x − x∗ ≥ ✈ỵ✐ ♠å✐ x ∈ C ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ tự (F, C) ữủ ợ t t✐➯♥ ✈➔♦ ♥➠♠ ✶✾✻✻ ❜ð✐ ●✳❏✳ ❍❛rt♠❛♥ ✈➔ ●✳ ❙t❛♠♣❛❝❝❤✐❛✱ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ✈✐➺❝ ❣✐↔✐ ❜➔✐ t♦→♥ ✤✐➲✉ ❦❤✐➸♥ tè✐ ÷✉ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❜✐➯♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❬✼❪✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❝â q✉❛♥ ❤➺ ♠➟t t❤✐➳t ✈ỵ✐ ♥❤✐➲✉ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥ ♥❤÷ ♠ỉ ❤➻♥❤ ❝➙♥ ❜➡♥❣ ♠↕♥❣ ❣✐❛♦ t❤ỉ♥❣✱ ❜➔✐ t tỹ t ỷ ỵ ✳ ◆➠♠ ✶✾✼✶✱ ▼✳ ❙✐❜♦♥② ❬✶✸❪ ✤➣ ①➨t ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ t➟♣ r➔♥❣ ❜✉ë❝ C ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû ✤ì♥ ✤✐➺✉✳ ❈ơ♥❣ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ■✳ ❨❛♠❛❞❛ ❬✶✽❪ ✤➣ ①➨t ❜➔✐ t♦→♥ ✈ỵ✐ t➟♣ C ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥ ✭tr÷í♥❣ ❤đ♣ r✐➯♥❣ ❦❤✐ C ❧➔ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû ✤ì♥ ✤✐➺✉✮✳ ◆❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❧➔ ♠ët ✤➲ t➔✐ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ t➼♥❤ ù♥❣ ❞ö♥❣ ❝õ❛ ❜➔✐ t♦→♥ ♥➔② tr♦♥❣ ♠ët sè ♥❣➔♥❤ ❦❤♦❛ ❤å❝✳ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ♠ð rë♥❣ t❤➔♥❤ ❝→❝ ❞↕♥❣ tê♥❣ q✉→t ❤ì♥ ♥❤÷ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✤❛ trà ✈ỵ✐ →♥❤ ①↕ F ❧➔ →♥❤ ①↕ ✤❛ trà✱ ❜➔✐ t♦→♥ ❝➙♥ ❜➡♥❣✱ ❜➔✐ t♦→♥ t➻♠ ✺ ✤✐➸♠ ❝❤✉♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✳ ✳ ✳ ▲✉➟♥ ✈➠♥ ♥❣❤✐➯♥ ❝ù✉ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ tr♦♥❣ ❜➔✐ ❜→♦ ❬✹❪✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ ✧❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ổ rt ữỡ ợ t t tỷ ❝❤✐➳✉✱ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✱ t♦→♥ tû ✤ì♥ ✤✐➺✉ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❝ị♥❣ ♠ët sè t➼♥❤ ❝❤➜t❀ tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt❀ ❣✐ỵ✐ t❤✐➺✉ ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ❞➝♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❈❤÷ì♥❣ ✷ ✧▼ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr ổ rt ữỡ ợ t ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❝ị♥❣ ♠ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥❀ ♠ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ✻ ❈❤÷ì♥❣ ✶ ❇➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤÷ì♥❣ ♥➔② ❣✐ỵ✐ t❤✐➺✉ ✈➲ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H✱ ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ❞➝♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð tê♥❣ ❤ñ♣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶✱ ✷✱ ✺✱ ✽✱ ✶✵✱ ✶✶✱ ✶✷✱ ✶✺✱ ✶✼❪✳ ✶✳✶ ▼ët sè t➼♥❤ ❝❤➜t ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤♦ H ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt tỹ C ởt t ỗ õ rộ H ỵ t ổ ữợ Ã, à ✈➔ ❝❤✉➞♥ t÷ì♥❣ ù♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ x = ✶✳✶✳✶ x, x ✈ỵ✐ ♠å✐ x ∈ H✳ ❙ü ❤ë✐ tö ②➳✉✱ ❤ë✐ tö ♠↕♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ✭①❡♠ ❬✶❪✮✳ ▼ët ❞➣② {xk } ⊂ H ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tử tử tợ x H ỵ ❤✐➺✉ xk → x∗ ✭t÷ì♥❣ ù♥❣ xk x∗ ✮✱ ♥➳✉ xk − x∗ → ✭t÷ì♥❣ ù♥❣ u, xk − x∗ → ✈ỵ✐ ♠å✐ u ∈ H✮ ❦❤✐ k → ∞✳ ▼ët ❞➣② {xk } ⊂ H ❤ë✐ tö ♠↕♥❤ ✤➳♥ x∗ t❤➻ ❝ơ♥❣ ❤ë✐ tư ②➳✉ ✤➳♥ x∗ ✱ ♥❤÷♥❣ ✤✐➲✉ ♥❣÷đ❝ ❧↕✐ ❦❤ỉ♥❣ ✤ó♥❣✳ ❚✉② ♥❤✐➯♥✱ t➼♥❤ ❝❤➜t ❑❛❞❡❝✕❑❧❡❡ ❝❤➾ r❛ r➡♥❣ xk → x∗ ❈❤♦ C = ∅, C ⊂ H✳ ✈➔ xk x∗ =⇒ xk → x∗ ✷✶ ❙û ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ✭✶✳✶✵✮ ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✭✶✳✷✵✮ ❝ơ♥❣ ❝❤➼♥❤ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✶✳✶✾✮ ✈ỵ✐ A(x) = ϕ(x) ❝â t➼♥❤ ❝❤➜t 2✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ 2✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ C ✳ ❑➳t q✉↔ t➼♥❤ t♦→♥ tr➯♥ ▼❆❚▲❆❇ ✤÷đ❝ ❝❤♦ tr♦♥❣ ❝→❝ ❇↔♥❣ ✶✳✶✕✶✳✸✳ ◆❤➟♥ ①➨t ✶✳✷✳✶✷✳ ❚ø ❇↔♥❣ ✶✳✶✕✶✳✸ ♥❤➟♥ t❤➜② ✈ỵ✐ ①➜♣ ①➾ ❜❛♥ ✤➛✉ ợ sỹ ỹ t số ũ ❤đ♣✱ t❛ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ ♥❣❤✐➺♠ ①➜♣ ①➾ ❦❤→ tèt ú t s ữợ ❇↔♥❣ ✶✳✶✿ ❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ x0 = (5, 5, 5)T ∈ R3 ✱ xk ❙❛✐ sè ✭ k ✭sè ❧➛♥ ❧➦♣✮ ✶✵ ✺✵ (1.0727, 2.0545, 3.0364)T T (1.0031, 2.0024, 3.016) xk − x∗ ✮ ❚✐♠❡ ✵✳✵✾✼✾ ✵✳✵✼✸s ✵✳✵✵✹✷ ✵✳✵✸✶s (1.0008, 2.0006, 3.0004)T ✵✳✵✵✶✶ ✵✳✵✷✼s ✺✵✵ (1.0000, 2.0000, 3.0000)T ✹✳✷✾✾✺❡✲✵✺ ✵✳✾✽✻s T ✶✳✵✼✻✵❡✲✵✺ ✶✳✵✵✻s (1.0000, 2.0000, 3.0000) ❇↔♥❣ ✶✳✷✿ ❇↔♥❣ t➼♥❤ t♦→♥ ✈ỵ✐ ✭sè ❧➛♥ ❧➦♣✮ x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 2) xk ❙❛✐ sè ✭ xk − x∗ ✮ ❚✐♠❡ ✶✵ (0.6182, 0.8727, 2.4000)T ✶✳✸✸✷✾ ✵✳✵✺✾s ✺✵ T ✵✳✵✺✼✺ ✵✳✵✸✼s ✶✵✵ (0.9958, 1.9877, 2.9935)T ✵✳✵✶✹✺ ✵✳✵✸✹s ✺✵✵ (0.9998, 1.9995, 2.9999)T ✺✳✸✷✽✹❡✲✵✹ ✶✳✵✹✺s T ✶✳✸✸✸✹❡✲✵✹ ✶✳✶✸✸s ✶✵✵✵ (0.9835, 1.9514, 2.9741) (1.0000, 1.9999, 3.0000) t t ợ k = 1/(k + 2) ✶✵✵ ✶✵✵✵ k ❝❤å♥ ✭sè ❧➛♥ ❧➦♣✮ x0 = (−20, −60, −10)T ∈ R3 ✱ µ = 1/(k + 4) xk ❙❛✐ sè ✭ xk − x∗ ✮ ❚✐♠❡ ✶✵ (−0.6154, −2.7692, 2.0000)T ✺✳✶✸✸✼ ✵✳✵✶✸s ✺✵ (0.9086, 1.7300, 2.9434)T ✵✳✷✾✵✻ ✵✳✵✸✼s ✶✵✵ (0.9760, 1.9292, 2.9852) T ✵✳✵✼✻✷ ✵✳✵✽✶s ✺✵✵ (0.9990, 1.9971, 2.9994)T ✵✳✵✵✸✷ ✶✳✵✼✼s ✼✳✾✻✽✼❡✲✵✹ ✶✳✶✸✽s ✶✵✵✵ (0.9997, 1.9993, 2.9998) T ✷✷ ❈❤÷ì♥❣ ✷ ▼ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ❝❤✐➳✉ ❣✐↔✐ ♠ët ❧ỵ♣ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❇❱■(F, G, C) tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✈ỵ✐ ❣✐↔ t❤✐➳t →♥❤ ①↕ F ❧➔ β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤✱ L✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ ✈➔ →♥❤ ①↕ G ❧➔ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝ tr➯♥ C ✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ♠ư❝✳ ▼ư❝ ✷✳✶ tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✳ ▼ư❝ ✷✳✷ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✈➔ sü ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ ✈✐➳t tr➯♥ ❝ì sð ❜➔✐ ❜→♦ ❬✹❪✳ ✷✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✷✳✶✳✶ ❇➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H✱ ①➨t →♥❤ ①↕ F, G : C → H✳ ❇➔✐ t t tự ỵ ❧➔ ❇❱■(F, G, C)✱ ❧➔ ❜➔✐ t♦→♥ t➻♠ x∗ ∈ S(G,C) t❤ä❛ ♠➣♥ F (x∗ ), x − x∗ ≥ ∀x ∈ S(G,C) , ð ✤➙② S(G,C) ❧➔ t➟♣ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ t➻♠ y ∗ ∈ C t❤ä❛ ♠➣♥ G(y ∗ ), x − y ∗ ≥ ∀x ∈ C ✭✷✳✶✮ ✷✸ ❚❛ ỵ t t (F, G, C) ❧➔ Ω ❇➔✐ t♦→♥ ❇❱■(F, G, C) ✤➣ ✤÷đ❝ ♥❤✐➲✉ t→❝ ❣✐↔ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✱ ✤➦❝ ❜✐➺t ❧➔ ✈✐➺❝ ①➙② ❞ü♥❣ ♠ët sè t❤✉➟t t♦→♥ ❣✐↔✐ ❞ü❛ tr➯♥ t➼♥❤ ✤ì♥ ✤✐➺✉ ❝õ❛ ❝→❝ →♥❤ ①↕ ❣✐→ F ✈➔ G✳ ✷✳✶✳✷ ▼ët sè ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥ ❚❤ỉ♥❣ t❤÷í♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❜➔✐ t♦→♥ tè✐ ÷✉ ❤❛✐ ❝➜♣ ♥â✐ ❝❤✉♥❣ ✈➔ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ♥â✐ r✐➯♥❣✱ ♥❣÷í✐ t❛ q✉❛♥ t➙♠ ✤➳♥ ❝→❝ t❤✉➟t t♦→♥ ❣✐↔✐ ❝ơ♥❣ ♥❤÷ ✤✐➲✉ tỗ t t õ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❝❤ù❛ ✤ü♥❣ ♠ët sè ❧ỵ♣ ❝õ❛ ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ ❤❛✐ ❝➜♣✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❜➔✐ t♦→♥ t➻♠ ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ t➟♣ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ tự ổ t ỗ ❝➜♣ ✈➔ ❜➔✐ t♦→♥ t✉②➳♥ t➼♥❤ ❤❛✐ ❝➜♣✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✱ ❝â ♥❤✐➲✉ t→❝ ❣✐↔ ✤➣ ✤÷❛ r❛ t❤✉➟t t♦→♥ t➻♠ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ữợ trữớ ủ r ữ tr trữớ ủ f, g ỗ ✈✐✱ ❜➔✐ t♦→♥ ❇❱■(F, G, C) ✭✈ỵ✐ F = ∇f ✈➔ G = ∇g ✮ ❝â ❞↕♥❣ ❝õ❛ ❜➔✐ t♦→♥ ❝ü❝ t✐➸✉ ❤❛✐ ❝➜♣ ❬✶✹❪  min f (x) x ∈ ❛r❣♠✐♥{g(x) : x ∈ C} ❚r÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t F (x) = x ✈ỵ✐ ♠å✐ x ∈ C ✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❇❱■(F, G, C) ❝â ❞↕♥❣ ❜➔✐ t♦→♥ t➻♠ ❝❤✉➞♥ ♥❤ä ♥❤➜t ❝õ❛ t➟♣ ♥❣❤✐➺♠ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ s❛✉ t➻♠ x∗ ∈ C s❛♦ ❝❤♦ x∗ = PS(G,C) (0) ✭✷✳✷✮ ❚❤✉➟t t♦→♥ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ✤➸ ❣✐↔✐ t ữủ ợ t tr ✈ỵ✐ ❣✐↔ t❤✐➳t t➟♣ r➔♥❣ ❜✉ë❝ C ⊆ H ❧➔ t ỗ õ rộ G : C → H ❧➔ α✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝ ✈➔ S(G,C) = ∅✳ ❚❤✉➟t t♦→♥ ✷✹ ✤÷đ❝ tr➻♥❤ ❜➔② ♥❤÷ s❛✉✿    ❚➻♠ x0 ∈ C,   y k = PC (xk − λG(xk ) − αk xk ),    xk+1 = P [xk − λG(xk ) + µ(y k − xk )], ∀k ≥ C ❑❤✐ ✤â✱ ❞➣② {xk } ❤ë✐ tö ♠↕♥❤ x = PS(G,C) (0) ữợ ởt số ✤➦t ❧➯♥ t❤❛♠ sè✳ ❚❛ ♥❤➢❝ ❧↕✐ ♠ët sè ❜ê ✤➲ ✤÷đ❝ ❞ị♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ❝õ❛ tr t t ỵ (S) ❧➔ t➟♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ S ✱ tù❝ ❧➔ ❋✐①(S) = {x ∈ C : x = Sx} ❇ê ✤➲ ✷✳✶✳✶ ✭①❡♠ ❬✻❪✮✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt tỹ H C t ỗ õ ré♥❣ ✈➔ S : C → H ❧➔ →♥❤ ①↕ ❦❤æ♥❣ ❣✐➣♥✳ ❑❤✐ ✤â✱ ♥➳✉ ❋✐①(S) = ∅✱ t❤➻ I S I ỗ t tr H ❧➔ ♥û❛ ✤â♥❣ t↕✐ y ∈ H tù❝ ❧➔✱ ✈ỵ✐ ❜➜t ❦ý ❞➣② {xk } t❤✉ë❝ C ❤ë✐ tö ②➳✉ ✤➳♥ ✤✐➸♠ x ¯ ∈ C ✈➔ ❞➣② {(I − S)(xk )} ❤ë✐ tö ♠↕♥❤ ✤➳♥ y ✱ t❛ ❝â (I − S)(¯ x) = y ✳ ❇ê ✤➲ ✷✳✶✳✷ ✭①❡♠ ❬✶✻❪✱ ❇ê ✤➲ ✷✳✺✮✳ ●✐↔ sû {an } ❧➔ ❞➣② sè t❤ü❝ ❦❤æ♥❣ ➙♠ t❤ä❛ ♠➣♥ an+1 ≤ (1 − γn )an + δn , ∀n ≥ 0, ✈ỵ✐ {γn } ⊂ (0, 1) ✈➔ {δn } ❧➔ ♠ët ❞➣② tr♦♥❣ R t❤ä❛ ♠➣♥ ∞ γn = ∞, (a) n=0 (b) lim sup γδnn ≤ ❤♦➦❝ n→∞ ❑❤✐ ✤â✱ lim an = 0✳ ∞ |δn γn | < +∞✳ n=0 n→∞ ✷✳✶✳✸ ❚❤✉➟t t♦→♥ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❚❤✉➟t t♦→♥ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❬✾❪ ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❤ú✉ ❤✐➺✉ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ✈ỵ✐ →♥❤ ①↕ ❣✐→ ✤ì♥ ✤✐➺✉ ✈➔ ❧✐➯♥ tư❝ ✷✺ ▲✐♣s❝❤✐t③✳ ●➛♥ ✤➙②✱ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✈✐➺❝ ①➙② ❞ü♥❣ t❤✉➟t t♦→♥ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣✱ t→❝ ❣✐↔ ❝õ❛ ❜➔✐ ❜→♦ ❬✸❪ ✤➣ →♣ ❞ư♥❣ t❤✉➟t t♦→♥ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ❜➡♥❣ ❝→❝❤ ①➙② ❞ü♥❣ ❝→❝ ❞➣② ❧➦♣ ♥❤÷ s❛✉✳ ❚❤✉➟t t♦→♥ ✷✳✶✳✸ ✭❬✸❪✱ ❚❤✉➟t t♦→♥ ✷✳✷✮✳ ❈❤♦ k = 0, x0 ∈ H, < λ ≤ 2β ✱ L21 ❝→❝ ❞➣② sè ❞÷ì♥❣ {δk }, {λk }, {αk }✱ {βk }, {γk } ✈➔ { k } t❤ä❛ ♠➣♥    {αk } ⊂ [m, n] ✈ỵ✐ m, n ∈ (0, 1), λk ≤ L12 ∀k ≥ 0,    ∞  lim δk = 0, k < ∞, < lim inf k→∞ βk < lim supk→∞ βk < 1, k→∞ k=0   ∞    + β + γ = 1, ∀k ≥ 0, lim = 0,  k k k k k = k k=0 ữợ xk t ❞ø♥❣✳ ◆❣÷đ❝ ❧↕✐✱ t➼♥❤ y k = PC (xk − λk G(xk )) ✈➔ z k = PC (xk − k G(y k )) ữợ ỏ tr j = 0, 1, ✳ ❚➼♥❤    xk,0 = z k − λF (z k ),   y k,j = PC (xk,j − δj G(xk,j )),    xk,j+1 = xk,0 + β xk,j + γ P (xk,j − δ G(y k,j )) j j j C j ❚➻♠ hk t❤ä❛ ♠➣♥ hk − lim xk,j ≤ j→∞ k ✈➔ ✤➦t xk+1 = αk xk + (1 k )hk ữợ ữủ ❧↕✐✱ t❤❛② k ❜ð✐ k + ✈➔ q✉❛② ❧↕✐ ữợ ỹ tử t t ữủ tr ỵ s ỵ ỵ C t ỗ õ ❦❤→❝ ré♥❣ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H✳ ●✐↔ sû →♥❤ ①↕ F : C → H ❧➔ β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ L1 ✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ C ✈➔ →♥❤ ①↕ G : C → H ❧➔ ✤ì♥ ✤✐➺✉ ✈➔ L2 ✲❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③ tr➯♥ C ✳ ❑❤✐ ✤â✱ ❝→❝ ❞➣② {xk }, {y k } ✈➔ {z k } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❚❤✉➟t t♦→♥ ✷✳✶✳✸ ❤ë✐ tư ♠↕♥❤ ✤➳♥ x∗ ❧➔ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❇➔✐ t♦→♥ ❇❱■(F, G, C)✳ ❍ì♥ ♥ú❛✱ t❛ ❝â x∗ = lim PS(G,C) (xk ) k→∞ ✷✻ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ✷✳✷✳✶ ▼æ t↔ ♣❤÷ì♥❣ ♣❤→♣ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ tr➻♥❤ ❜➔② t❤✉➟t t♦→♥ ❝❤✐➳✉ tr♦♥❣ ❬✹❪ ✤➸ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❇❱■(F, G, C) ❞ü❛ tr➯♥ ❝ì sð ❦➳t ❤ñ♣ ❣✐ú❛ t❤✉➟t t♦→♥ ❝❤✐➳✉ ✤↕♦ ❤➔♠ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❦❤ỉ♥❣ ❣✐➣♥✳ t t ỗ ữợ ữợ ỷ tt t♦→♥ ❝❤✐➳✉ ✤↕♦ ❤➔♠ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❱■(G, C) ✈➔ t➼♥❤ ❞➣② ❧➦♣ xk+1 = PC (xk − λG(xk )) (k = 0, 1, ) ✈ỵ✐ λ > ✈➔ x0 ∈ C ữợ ỷ ỵ t ✤ë♥❣ ❝õ❛ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ t➻♠ ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t ❝õ❛ →♥❤ ①↕ ❝♦ Tλ = I − àF ợ I ỗ t (0, 2β ) ✈➔ λ ∈ (0, 1]✳ L2 ●✐↔ t❤✐➳t ✷✳✷✳✶✳ ●✐↔ sû →♥❤ ①↕ F ✈➔ G t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭❈✶✮ G ❧➔ →♥❤ ①↕ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝ tr➯♥ H❀ ✭❈✷✮ F ❧➔ →♥❤ ①↕ β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈➔ L✲❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ tr➯♥ C ❀ ✭❈✸✮ ❚➟♣ ♥❣❤✐➺♠ Ω ❝õ❛ ❜➔✐ t♦→♥ ❇❱■(F, G, C) ❦❤→❝ ré♥❣✳ ❑❤✐ ✤â✱ ❝→❝ ❞➣② ❧➦♣ ❝õ❛ t❤✉➟t t♦→♥ ✤÷đ❝ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ♥❤÷ s❛✉✳ ❚❤✉➟t t♦→♥ ✷✳✷✳✷ ✭①❡♠ ❬✹❪✮✳ ❈❤å♥ x0 ∈ C, k = số ữỡ {k }, , tọ ♠➣♥   0 < αk ≤ min{1, τ1 }, τ = −   lim αk = 0, lim k→∞ k→∞ αk+1 − αk − µ(2β − µL2 ), ∞ αk = ∞, < λ ≤ 2η, < µ < = 0, k=0 ữợ tự k, (k = 0, 1, 2, ) õ xk tỹ ữợ s ữợ y k = PC (xk G(xk )) L2 ữợ xk+1 = y k − µαk F (y k )✳ ◆➳✉ xk+1 = xk ✱ t❤➻ t❤✉➟t t♦→♥ ❞ø♥❣✱ xk ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❇❱■(F, G, C)✳ ◆❣÷đ❝ ❧↕✐✱ s ữợ tự k ợ k ữủ t ❜ð✐ k + 1✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ F (x) = ✈ỵ✐ ♠å✐ x ∈ C ✱ ❞➣② ❧➦♣ {xk } tr♦♥❣ ❚❤✉➟t t♦→♥ ✷✳✷✳✷ ✤÷đ❝ ①→❝ ✤à♥❤ t❤ỉ♥❣ q✉❛ ❞➣② ❧➦♣ xk+1 = PC (xk − λG(xk )) ✷✳✷✳✷ ❙ü ❤ë✐ tö ❙ü ❤ë✐ tö ♠↕♥❤ ❝õ❛ ❚❤✉➟t t♦→♥ ữủ t tổ q ỵ s ỵ C ởt t ỗ õ rộ ởt ổ rt t❤ü❝ H✳ ●✐↔ sû →♥❤ ①↕ F : C → H ✈➔ G : H → H t❤ä❛ ♠➣♥ ❝→❝ ❣✐↔ t❤✐➳t ✭❈✶✮✕✭❈✸✮✳ ❑❤✐ ✤â✱ ❝→❝ ❞➣② {xk } ✈➔ {y k } ①→❝ ✤à♥❤ ❜ð✐ ❚❤✉➟t t♦→♥ ✷✳✷✳✷ ❤ë✐ tư ♠↕♥❤ tỵ✐ ♥❣❤✐➺♠ ❞✉② ♥❤➜t x∗ ∈ Ω✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✮ ❝õ❛ ❚❤✉➟t t♦→♥ ✷✳✷✳✷✱ t❛ ①➙② ❞ü♥❣ →♥❤ ①↕ Sk : H → H ♥❤÷ s❛✉ Sk (x) = P rC (x − λG(x)) − µαk F [P rC (x − λG(x))], ∀x ∈ H ❚❤❡♦ ❣✐↔ t❤✐➳t✱ G ❧➔ →♥❤ ①↕ η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝✳ ❇➡♥❣ ✈✐➺❝ sû ❞ư♥❣ t➼♥❤ ❝❤➜t ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉ ❝ò♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✮✱ ❦❤✐ ✤â ✈ỵ✐ ♠å✐ x, y ∈ H✱ t❛ ❝â PC (x − λG(x)) − PC (y − λG(y)) ≤ x − λG(x) − y + λG(y) = x−y 2 + λ2 G(x) − G(y) − 2λ x − y, G(x) − G(y) ≤ x−y + λ(λ − 2η) G(x) − G(y) ≤ x − y 2 ✭✷✳✹✮ ✷✽ ❑➳t ❤đ♣ ✭✷✳✹✮ ✈ỵ✐ ❇ê ✤➲ ✶✳✶✳✶✵✱ t❛ ✤÷đ❝ Sk (x) − Sk (y) = PC (x − λG(x)) − µαk F [PC (x − λG(x))] − PC (y − λG(y)) + µαk F [PC (y − λG(y))] ✭✷✳✺✮ ≤(1 − αk τ ) x − y , ợ = à(2 µL2 ) ∈ (0, 1]✳ ❉♦ ✤â✱ Sk ❧➔ →♥❤ tr H ỵ tỗ t t k tọ Sk (ξ k ) = ξ k ✳ ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ x ˆ ∈ S(G, C)✱ ✤➦t Cˆ = x∈H: x − xˆ ≤ µ F (ˆ x) τ , ❦➳t ❤đ♣ ✈ỵ✐ t➼♥❤ ❝❤➜t ❦❤ỉ♥❣ ❣✐➣♥ ❝õ❛ ♣❤➨♣ ❝❤✐➳✉✱ s✉② r❛ →♥❤ ①↕ Sk PCˆ ❧➔ →♥❤ ①↕ tr H tỗ t t z k t❤ä❛ ♠➣♥ Sk [PCˆ (z k )] = z k ✳ ✣➦t z¯k = PCˆ (z k )✱ tø ✭✷✳✺✮ ✈➔ ❝→❝❤ ✤➦t →♥❤ ①↕ Sk ✱ t❛ ❝â z k − xˆ = Sk (¯ z k ) − xˆ ≤ Sk (¯ z k ) − Sk (ˆ x) + Sk (ˆ x) − xˆ = Sk (¯ z k ) − Sk (ˆ x) + Sk (ˆ x) − PC (ˆ x − αk G(ˆ x)) ≤(1 − αk τ ) z¯k − xˆ + µαk F [PC (ˆ x − αk G(ˆ x))] µ F (ˆ x) ≤(1 − αk τ ) + µαk F (ˆ x) τ µ F (ˆ x) = τ ˆ ✈➔ Sk [P ˆ (z k )] = Sk (z k ) = z k ✳ ❉♦ ✈➟②✱ ξ k = z k ∈ Cˆ ✳ ✣✐➲✉ ♥➔② ❝❤➾ r❛ r➡♥❣ z k ∈ C C ▼➦t ❦❤→❝✱ ✈ỵ✐ ❜➜t ❦ý ❞➣② ❝♦♥ {ξ ki } ❝õ❛ ❞➣② {ξ k } t❤ä❛ ♠➣♥ ξ ki ξ¯ ✈➔ lim αk = 0, k→∞ t❛ ❝â PC (ξ ki − λG(ξ ki )) − ξ ki = PC (ξ ki − λG(ξ ki )) − Ski (ξ ki ) =µαki F [PC (ξ ki − λG(ξ ki ))] →0 ❦❤✐ i → ∞ ✭✷✳✻✮ ✷✾ ❚❤❡♦ ✭✷✳✹✮✱ →♥❤ ①↕ PC (· − αk G(·)) ❧➔ ❦❤æ♥❣ ❣✐➣♥ tr➯♥ H✱ ❦➳t ❤đ♣ ✈ỵ✐ ❇ê ✤➲ ✷✳✶✳✶✱ ✭✷✳✻✮ ✈➔ ξ ki ¯ = ξ¯✳ ❱➟② ξ¯ ∈ S(G, C)✳ ξ¯✱ s✉② r❛ PC (ξ¯ − λG(ξ)) ❚✐➳♣ t❤❡♦✱ t❛ ❝❤ù♥❣ ♠✐♥❤ lim ξ kj = x∗ ∈ Ω✳ ❚❤➟t ✈➟②✱ ✤➦t j→∞ z¯k = PC (ξ k − λG(ξ k )), v ∗ = (µF − I)(x∗ ) ✈➔ v k = (µF − I)(¯ z k )✱ ð ✤➙② I ❧➔ ỗ t Skj ( kj ) = ξ kj ✈➔ x∗ = PC (x∗ − λG(x∗ )) ♥➯♥ t❛ ❝â (1 − αkj )(ξ kj − z¯kj ) + αkj (ξ kj + v kj ) = ✈➔ (1 − αkj )[I − PC (· − λG(·))](x∗ ) + αkj (x∗ + v ∗ ) = αkj (x∗ + v ∗ ) ❑❤✐ ✤â −αkj x∗ + v ∗ , ξ kj − x∗ =(1 − αkj ) ξ kj − x∗ − (¯ z kj − x∗ ), ξ kj − x∗ + αkj ξ kj − x∗ + v kj − v ∗ , ξ kj − x∗ ✭✷✳✼✮ ❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ❙❝❤✇❛r③✱ t❛ ❝â ξ kj − x∗ − (¯ z kj − x∗ ), ξ kj − x∗ ≥ ξ kj − x∗ − z¯kj − x∗ ≥ ξ kj − x∗ − ξ kj − x∗ ξ kj − x∗ ✭✷✳✽✮ =0, ✈➔ ξ kj − x∗ + v kj − v ∗ , ξ kj − x∗ ≥ ξ kj − x∗ − v kj − v ∗ ≥ ξ kj − x ∗ − (1 − τ ) ξ kj − x∗ ξ kj − x∗ =τ ξ kj − x∗ ❑➳t ❤đ♣ ✭✷✳✼✮✱ ✭✷✳✽✮ ✈➔ ✭✷✳✾✮✱ t❛ ✤÷đ❝ −τ ξ kj − x∗ ≥ x ∗ + v ∗ , ξ kj − x ∗ ) =µ F (x∗ ), ξ kj − x∗ =µ F (x∗ ), ξ kj − ξ¯ + µ F (x∗ ), ξ¯ − x∗ ≥µ F (x∗ ), ξ kj − ξ¯ ✭✷✳✾✮ ✸✵ ❱➟② τ ξ kj − x∗ ≤ µ F (x∗ ), ξ¯ − ξ kj ❈❤♦ j → ∞✱ ❞➣② {ξ kj } ❤ë✐ tử x õ tỗ t ởt ❞➣② ❝♦♥ {ξ kj } ❝õ❛ ❞➣② {ξ k } t❤ä❛ ♠➣♥ ≤ lim inf ξ k − x∗ ≤ lim sup ξ k − x∗ = lim ξ kj − x∗ = k→∞ j→∞ k→∞ ❱➟②✱ ❞➣② {ξ k } ❤ë✐ tö ♠↕♥❤ ✤➳♥ ✤✐➸♠ x∗ ∈ Ω✳ ▼➦t ❦❤→❝✱ t❤❡♦ ✭✷✳✺✮✱ t❛ ①➨t xk − ξ k ≤ xk − ξ k−1 + ξ k−1 − ξ k = Sk−1 (xk−1 ) − Sk−1 (ξ k−1 ) + ξ k−1 − ξ k ≤(1 − αk−1 τ ) xk−1 − ξ k−1 + ξ k−1 − ξ k ✭✷✳✶✵✮ ❍ì♥ ♥ú❛✱ t❤❡♦ ❇ê ✤➲ ✶✳✶✳✶✵✱ t❛ ❝â ξ k−1 − ξ k = Sk−1 (ξ k−1 ) − Sk (ξ k ) = (1 − αk )¯ z k − αk v k − (1 − αk−1 )¯ z k−1 + αk−1 v k−1 = (1 − αk )(¯ z k − z¯k−1 ) − αk (v k − v k−1 ) + (αk−1 − αk )(¯ z k−1 + v k−1 ) ≤(1 − αk ) z¯k − z¯k−1 + αk v k − v k−1 + |αk−1 − αk |µ F (¯ z k−1 ) ≤(1 − αk ) z¯k − z¯k−1 + αk − µ(2β − µL2 ) ξ k − ξ k−1 + |αk−1 − αk |µ F (¯ z k−1 ) ≤(1 − αk ) ξ k − ξ k−1 + αk − µ(2β − µL2 ) ξ k − ξ k−1 + |αk−1 − αk |µ F (¯ z k−1 ) ❱➟② αk τ ξ k−1 − ξ k ≤ |αk−1 − αk |µ F (¯ z k−1 ) ❙✉② r❛ k ξ −ξ k−1 µ|αk−1 − αk | F (¯ z k−1 ) ≤ αk τ ❚❤❛② ✭✷✳✶✶✮ ✈➔♦ ✭✷✳✶✵✮✱ t❛ ✤÷đ❝ xk − ξ k ≤ (1 − αk−1 τ ) xk−1 − ξ k−1 + µ|αk−1 − αk | F (¯ z k−1 ) αk τ ✭✷✳✶✶✮ ✸✶ ✣➦t µ|αk − αk+1 | F (¯ zk) δk = , k ≥ αk αk+1 τ ❉♦ ✤â xk − ξ k ≤ (1 − αk−1 τ ) xk−1 − ξ k−1 + αk−1 τ δk−1 , ∀k ≥ ❱➻ {F (¯ z k )} ❜à ❝❤➦♥✱ ❣✐↔ sû F (¯ z k ) ≤ K ✈ỵ✐ ♠å✐ k ≥ 0✱ t❛ ❝â µ|αk − αk+1 | F (¯ zk) lim δk = lim k→∞ k→∞ αk αk+1 τ µK 1 ≤ lim − τ k−→∞ αk+1 αk = ❉♦ ✤â✱ t❤❡♦ ❇ê ✤➲ ✷✳✶✳✷ s✉② r❛ lim xk − ξ k = 0✳ ▼➦t ❦❤→❝✱ t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ k→∞ tr➯♥✱ ❞➣② {ξ } ❤ë✐ tö ♠↕♥❤ ✤➳♥ ♥❣❤✐➺♠ x∗ ✱ s✉② r❛ ❞➣② {xk } ❝ơ♥❣ ❤ë✐ tư ♠↕♥❤ k ✤➳♥ ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ❜➔✐ t♦→♥ ❇❱■(F, G, C)✳ ❳➨t tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❦❤✐ F (x) = x ✈ỵ✐ ♠å✐ x ∈ H✳ ❚❛ t❤➜② F ❧➔ →♥❤ ①↕ L✲❧✐➯♥ tư❝ ▲✐♣s❝❤✐t③ ✈ỵ✐ ❤➺ sè L = ✈➔ β ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ✈ỵ✐ ❤➺ sè β = tr➯♥ H✳ ❑❤✐ ✤â✱ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ ❇❱■(F, G, C) ❝â ❞↕♥❣ ❜➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t tr➯♥ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ❍➺ q✉↔ ✷✳✷✳✹ ✭①❡♠ ❬✹❪✮✳ ❈❤♦ C t ỗ õ rộ ổ ❍✐❧❜❡rt t❤ü❝ H ✈➔ →♥❤ ①↕ G : H → H t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t η ✲✤ì♥ ✤✐➺✉ ♠↕♥❤ ♥❣÷đ❝✳ ợ < 2, < < 2✱ ❞➣② ❧➦♣ {xk } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐  y k = PC (xk − λG(xk )), xk+1 = (1 − µα )y k ✭✷✳✶✷✮ k ❈→❝ ❞➣② t❤❛♠ sè t❤ä❛ ♠➣♥   0 < αk ≤ min{1, }, τ = − |1 − µ|, τ ∞   lim αk = 0, lim αk+1 − α1k = 0, αk = ∞ k→∞ k→∞ k=0 ❑❤✐ ✤â✱ ❞➣② {x } ✈➔ {y } ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✷✮ ❤ë✐ tư ♠↕♥❤ ✤➳♥ ❝ị♥❣ ♠ët k ✤✐➸♠ x ˆ = PS(G,C) (0)✳ k ✸✷ ❑➳t ❧✉➟♥ ✣➲ t➔✐ ❧✉➟♥ ✈➠♥ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝ H ✳ ❈ư t❤➸✿ ✶✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt t❤ü❝❀ tr➻♥❤ ❜➔② ♠ët ❜➔✐ t♦→♥ t❤ü❝ t➳ ❞➝♥ ✤➳♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ✷✳ ❚r➻♥❤ ❜➔② ♠è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ợ t ữỡ tr t tỷ ❜➔✐ t♦→♥ tè✐ ÷✉✱ ❜➔✐ t♦→♥ ✤✐➸♠ ❜➜t ✤ë♥❣❀ tø ✤â tr➻♥❤ ❜➔② ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❧➦♣ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❞ü❛ tr➯♥ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➸♠ ❜➜t ✤ë♥❣ ✈➔ t➼♥❤ t♦→♥ ✈➼ ❞ư sè ♠✐♥❤ ❤å❛✳ ✸✳ ●✐ỵ✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣❀ ♠ỉ t↔ ♣❤÷ì♥❣ ♣❤→♣ ✤↕♦ ❤➔♠ t➠♥❣ ❝÷í♥❣ ✈➔ ♠ët ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ❣✐↔✐ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥ ❤❛✐ ❝➜♣❀ ❝❤ù♥❣ ♠✐♥❤ sü ❤ë✐ tư ♠↕♥❤ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✐➳✉ ✈➔ ①➨t tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❧➔ ❜➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝â ❝❤✉➞♥ ♥❤ä ♥❤➜t tr➯♥ t➟♣ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜✐➳♥ ♣❤➙♥✳ ✸✸ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❍♦➔♥❣ ❚ö②✱ ❍➔♠ t❤ü❝ ✈➔ ●✐↔✐ t➼❝❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍➔ ◆ë✐✱ ✷✵✵✺✳ ❚✐➳♥❣ ❆♥❤ ❬✷❪ ❘✳P✳ ❆❣❛r✇❛❧✱ ❉✳ ❖✬❘❡❣❛♥✱ ❉✳❘✳ ❙❛❤✉ ✭✷✵✵✾✮✱ ❋✐①❡❞ P♦✐♥t ❚❤❡♦r② ❢♦r ▲✐♣s❝❤✐t③✐❛♥✲t②♣❡ ▼❛♣♣✐♥❣s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✳ ❬✸❪ P✳◆✳ ❆♥❤ ✭✷✵✶✷✮✱ ✧❆ ♥❡✇ ❡①tr❛❣r❛❞✐❡♥t ✐t❡r❛t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ❜✐❧❡✈❡❧ ✈❛r✐✲ ❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✳✱ ✸✼✱ ♣♣✳ ✾✺✲✶✵✼✳ ❬✹❪ ❚✳❚✳❍✳ ❆♥❤✱ ▲✳❇✳ ▲♦♥❣✱ ❚✳❱✳ ❆♥❤ ✭✷✵✶✹✮✱ ✧❆ ♣r♦❥❡❝t✐♦♥ ♠❡t❤♦❞ ❢♦r ❜✐❧❡✈❡❧ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ❏✳ ■♥❡q✉❛❧✳ ❆♣♣❧✳✱ ✷✵✶✹✿✷✵✺✳ ❬✺❪ ❍✳❍✳ ❇❛✉s❝❤❦❡✱ P✳▲ ❈♦♠❜❡tt❡s ✭✷✵✶✵✮✱ ❈♦♥✈❡① ❛♥❛❧②s✐s ❛♥❞ ♠♦♥♦t♦♥❡ ♦♣✲ ❡r❛t♦r t❤❡♦r② ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡s✱ ❙♣r✐♥❣❡r✳ ❬✻❪ ❑✳ ●♦❡❜❡❧✱ ❲✳❆✳ ❑✐r❦ ✭✶✾✾✵✮✱ ❚♦♣✐❝s ♦♥ ♠❡tr✐❝ ❢✐①❡❞ ♣♦✐♥t t❤❡♦r②✱ ❈❛♠✲ ❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✱ ❊♥❣❧❛♥❞✳ ❬✼❪ P✳❚✳ ❍❛r❦❡r✱ ❏✳❙✳ P❛♥❣ ✭✶✾✾✵✮✱ ✧❆ ❞❛♠♣❡❞✲◆❡✇t♦♥ ♠❡t❤♦❞ ❢♦r t❤❡ ❧✐♥❡❛r ❝♦♠♣❧❡♠❡♥t❛r✐t② ♣r♦❜❧❡♠✧✱ ▲❡❝t✉r❡s ✐♥ ❆♣♣❧✳ ▼❛t❤✳✱ ✷✻✱ ♣♣✳ ✷✻✺✲✷✽✹✳ ❬✽❪ ■✳❱✳ ❑♦♥♥♦✈ ✭✷✵✵✶✮✱ ❈♦♠❜✐♥❡❞ ❘❡❧❛①❛t✐♦♥ ▼❡t❤♦❞s ❢♦r ❱❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✲ ✐t✐❡s✱ ❙♣r✐♥❣❡r ❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ●❡r♠❛♥②✳ ✸✹ ❬✾❪ ●✳▼✳ ❑♦r♣❡❧❡✈✐❝❤ ✭✶✾✼✻✮✱ ✧❆♥ ❡①tr❛❣r❛❞✐❡♥t ♠❡t❤♦❞ ❢♦r ❢✐♥❞✐♥❣ s❛❞❞❧❡ ♣♦✐♥ts ❛♥❞ ♦t❤❡r ♣r♦❜❧❡♠s✧✱ ❊❦♦♥♦♠✐❦❛ ✐ ▼❛t❡♠❛t✐❝❤❡s❦✐❡ ▼❡t♦❞②✱✶✷✱ ♣♣✳ ✼✹✼✲✼✺✻✳ ❬✶✵❪ P✳❊✳ ▼❛✐♥❣➨ ✭✷✵✵✽✮✱ ✧❙tr♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♣r♦❥❡❝t❡❞ s✉❜❣r❛❞✐❡♥t ♠❡t❤♦❞s ❢♦r ♥♦♥s♠♦♦t❤ ❛♥❞ ♥♦♥str✐❝t❧② ❝♦♥✈❡① ♠✐♥✐♠✐③❛t✐♦♥✧✱ ❙❡t✲❱❛❧✳ ❆♥❛❧✳✱ ✶✻✱ ♣♣✳ ✽✾✾✲✾✶✷✳ ❬✶✶❪ P✳❊✳ ▼❛✐♥❣➨ ✭✷✵✶✵✮✱ ✧Pr♦❥❡❝t❡❞ s✉❜❣r❛❞✐❡♥t t❡❝❤♥✐q✉❡s ❛♥❞ ✈✐s❝♦s✐t② ♠❡t❤✲ ♦❞s ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ✇✐t❤ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ❝♦♥str❛✐♥ts✧✱ ❊✉r✳ ❏✳ ❖♣❡r✳ ❘❡s✳ ✷✵✺✱ ♣♣✳ ✺✵✶✲✺✵✻✳ ❬✶✷❪ ▼✳❆✳ ◆♦♦r ✭✶✾✾✶✮✱ ✧❆♥ ✐t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠ ❢♦r ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s✧✱ ❏✳ ▼❛t❤❡♠❛t✐❝s ❆♥❛❧✳ ❆♣♣❧✳✱ ✶✺✽✱ ✹✹✽✕✹✺✺✳ ❬✶✸❪ ▼✳ ❙✐❜♦♥② ✭✶✾✼✶✮✱ ✧❙✉r ■✬❛♣♣r♦①✐♠❛t✐♦♥ ❞✬➨q✉❛t✐♦♥ ❡t ✐♥➨q✉❛t✐♦♥s ❛✉① ❞➨r✐✈➨❡s ♣❛rt✐❡❧❧❡s ♥♦♥❧✐♥➨❛✐r❡s ❞❡ t②♣❡ ♠♦♥♦t♦♥❡✧✱ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣✳✱ ✸✹✱ ♣♣✳ ✺✵✷✲✺✻✹✳ ❬✶✹❪ ▼✳ ❙♦❧♦❞♦✈ ✭✷✵✵✼✮✱ ✧❆♥ ❡①♣❧✐❝✐t ❞❡s❝❡♥t ♠❡t❤♦❞ ❢♦r ❜✐❧❡✈❡❧ ❝♦♥✈❡① ♦♣t✐✲ ♠✐③❛t✐♦♥✧✱ ❏✳ ❈♦♥✈❡① ❆♥❛❧✳✱ ✶✹✱ ♣♣✳ ✷✷✼✲✷✸✼✳ ❬✶✺❪ ❍✳ ❚✉② ✭✶✾✾✼✮✱ ❈♦♥✈❡① ❛♥❛❧②s✐s ❛♥❞ ❣❧♦❜❛❧ ♦♣t✐♠✐③❛t✐♦♥✱ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ❉♦r❞r❡❝❤t✳ ❬✶✻❪ ❍✳❑✳ ❳✉ ✭✷✵✵✷✮✱ ✧■t❡r❛t✐✈❡ ❛❧❣♦r✐t❤♠s ❢♦r ♥♦♥❧✐♥❡r ♦♣❡r❛t♦rs✧✱ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳✱ ✻✻✱ ♣♣✳ ✷✹✵✲✷✺✻✳ ❬✶✼❪ ■✳ ❨❛♠❛❞❛ ✭✷✵✵✶✮✱ ❚❤❡ ❤②❜r✐❞ st❡❡♣❡st ❞❡s❝❡♥t ♠❡t❤♦❞ ❢♦r t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ♣r♦❜❧❡♠ ♦✈❡r t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❢✐①❡❞ ♣♦✐♥t s❡ts ♦❢ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✱ ■♥ ✐♥❤❡r❡♥t❧② ♣❛r❛❧❧❡❧ ❛❧❣♦r✐t❤♠ ❢♦r ❢❡❛s✐❜✐❧✐t② ❛♥❞ ♦♣t✐♠✐③❛t✐♦♥ ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s ❡❞✐t❡❞ ❜②✿ ❉✳ ❇✉t♥❛r✐✉✱ ❨✳ ❈❡♥s♦r✱ ❛♥❞ ❙✳ ❘❡✐❝❤✱ ❊❧s❡✈✐❡r✳✱ ✹✼✸ ✲ ✺✵✹✳ ❬✶✽❪ ■✳ ❨❛♠❛❞❛✱ ◆✳ ❖❣✉r❛ ✭✷✵✵✺✮✱ ✧❍②❜r✐❞ st❡❡♣❡st ❞❡s❝❡♥t ♠❡t❤♦❞ ❢♦r t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ♣r♦❜❧❡♠ ♦✈❡r t❤❡ ❢✐①❡❞ ♣♦✐♥t s❡t ♦❢ ❝❡rt❛✐♥ q✉❛s✐✲ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣s✧✱ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠✳✱ ✷✺✱ ♣♣✳ ✻✶✾✲✻✺✺✳ ✸✺ ❬✶✾❪ ❨✳ ❨❛♦✱ ●✳ ▼❛r✐♥♦✱ ▲✳ ▼✉❣❧✐❛ ✭✷✵✶✹✮✱ ✧❆ ♠♦❞✐❢✐❡❞ ❑♦r♣❡❧❡✈✐❝❤✬s ♠❡t❤♦❞ ❝♦♥✈❡r❣❡♥t t♦ t❤❡ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ♦❢ ❛ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②✧✱ ❖♣✲ t✐♠✐③❛t✐♦♥✱ ✻✸✱ ♣♣✳ ✺✺✾✲✺✻✾✳